I honestly do not mind some restrictions, especially if it stops parents from calling their children obscene words (it happens). But there should still be quite a lot of freedom to name your child whatever you want. Picking from a small list seems restricting to me, or in some countries the name can only originate there.

The way I see it, your child isn't your actual property so the capital YOUR makes me uncomfortable. Children are their own people, and their parents just look after them until they are old enough to look after themselves. Of course kids can't choose their own names (or at least they need one to use until they can choose their own names) so the parents have to do that and one would hope that all parents, or legal guardians, pick a name with the best interests of the child at heart. Sometimes, sadly, parents don't have the best interests of their child at heart. Whilst naming your kid Traylyr Parq is not anywhere close to unacceptable behaviour from parents where the government does step in (e.g. neglect), I would argue that it would be pretty unkind in America. In Iceland it would be even more so; that combination of letters simply isn't comprehensible in an Icelandic language context.
Of course children don't belong to the government either, but I suppose the thinking is that they are likely to be more objective about it.

I honestly do not mind some restrictions, especially if it stops parents from calling their children obscene words (it happens). But there should still be quite a lot of freedom to name your child whatever you want. Picking from a small list seems restricting to me, or in some countries the name can only originate there.

Agreed. I think Iceland Government got sued because they refused name Blaer and forced the girl to live as Girl, only because it wasn't on the list.

I think people got too far in attempt to shield children from jokes and uniqueness. Not every unusual name is going to be a trouble and not every joke or comment is abuse.

Agreed. I think Iceland Government got sued because they refused name Blaer and forced the girl to live as Girl, only because it wasn't on the list.

Er... kind of. The reporting of this issue was not great - the barrier was that a name is not supposed to be on the list of boys' names and and the list of girls' names. Blær was on the list, but only as a boy's name - it is still basically considered a male name. However, it actually fulfilled all the requirements to be on the list as a girl's name as well, so now it is on the list for both sexes. Blær Bjarkardóttir is the only female who has it as a first name. It's not a common name either way, only 7 people in the entire country are named that. Around a hundred people have it as a middle name, but overwhelmingly men and boys.

Edit: Correction, there are two females who have Blær as a first name and five males.